![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tpostpos2 | Structured version Visualization version GIF version |
Description: Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
tpostpos2 | ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpostpos 8228 | . 2 ⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V)) | |
2 | relrelss 6270 | . . . 4 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) ↔ 𝐹 ⊆ ((V × V) × V)) | |
3 | ssun1 4172 | . . . . . 6 ⊢ (V × V) ⊆ ((V × V) ∪ {∅}) | |
4 | xpss1 5695 | . . . . . 6 ⊢ ((V × V) ⊆ ((V × V) ∪ {∅}) → ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V) |
6 | sstr 3990 | . . . . 5 ⊢ ((𝐹 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) | |
7 | 5, 6 | mpan2 690 | . . . 4 ⊢ (𝐹 ⊆ ((V × V) × V) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
8 | 2, 7 | sylbi 216 | . . 3 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
9 | df-ss 3965 | . . 3 ⊢ (𝐹 ⊆ (((V × V) ∪ {∅}) × V) ↔ (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) | |
10 | 8, 9 | sylib 217 | . 2 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) |
11 | 1, 10 | eqtrid 2785 | 1 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Vcvv 3475 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {csn 4628 × cxp 5674 dom cdm 5676 Rel wrel 5681 tpos ctpos 8207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-fv 6549 df-tpos 8208 |
This theorem is referenced by: 2oppchomf 17667 mattpostpos 21948 opprabs 32585 |
Copyright terms: Public domain | W3C validator |